Some aspects of profinite group theory
Dan Segal
综述了投射有限群的最新结果，以及投射有限群理论起主导作用的无限群和有限群的结果。
A survey of recent results about profinite groups, and results about infinite and finite groups where the theory of profinite groups plays a leading role.
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Closed subgroups of free profinite monoids are projective profinite groups
证明了自由凸有限幺半群的闭子群类恰是投射投射有限群类。特别是，阿尔梅达提出的与最小符号动力系统相关的投射有限群是投射的。我们的结果回答了Lubotzky在阿尔梅达的演讲中提出的一个问题，在投射有限群和应用领域研讨会，卡尔顿大学，2005年8月。我们还证明了自由profinite幺半群的任何有限子半群由幂等元组成。
Profinite extensions of centralizers and the profinite completion of limit groups
引入并研究了一类通过中心化子的扩张定义的投射有限群，这类群类似于广泛研究的完全剩余生成的自由群，即极限群（在Z. Sela）。从极限群的profinite完备化属于这类这一事实出发，得到了极限群的群论结构和同调性质的结果。
In this paper we investigate some properties of the Burnside ring of a profinite group as defined in \cite{ds}. We introduce the notion of the crossed Burnside ring of a profinite FC-group, and generalise some results from finite to profinite (FC-)groups. In our investigations, we also obtain results on profinite FC-groups which may be of independent interest.
Groups of profinite type and profinite rigidity
本文研究 profinite 型群（可实现为 Galois 群的群）及其 profinite 刚性（唯一 profinite 拓扑），探讨阿贝尔群与群扩张的性质，并联系上同调比较映射的单射性与满射性。
We generalise to profinite groups some of our previous results on the cohomology of pro-p groups of bounded sectional p-rank.
We study homological approximations of the profinite completion of a limit group (see Thm.~A) and obtain the analogous of Bridson and Howie's Theorem for the profinite completion of a non-abelian limit group (see Thm.~B).
An uncountable family of 3-generated groups with isomorphic profinite completions
构造了一类具有同构profinite完备化的不可数3-生成剩余有限恰好无限群.我们还表明，字的增长率是不是一个profinite属性。
Fundaments of epimorphisms of profinite groups
本文提出并发展了一种新理论，通过不可分解满态射来刻画Profinite群的满态射。
Positively finitely related profinite groups
我们定义并研究了正有限相关（PFR）投射有限群类。正有限相关性是投射有限群的一个概率性质，它为定义投射有限群的更高有限性提供了第一步，推广了Avinoam Mann引入的正有限生成群.我们证明了PFR群的许多渐近性质，例如我们证明了：一个完全表示的投射有限群是PFR当且仅当它在有限域上一致地至多有指数表示增长（换句话说：完备的群代数有多项式极大理想增长）。从这些刻画中，我们推出了PFR投射有限群的几个结构结果。
We show that every closed normal subgroup of infinite index in a profinite surface group $Γ$ is contained in a semi-free profinite normal subgroup of $Γ$. This answers a question of Bary-Soroker, Stevenson, and Zalesskii.
Finitely generated (non-abelian) free metabelian pro-p groups, and wreath products of f.g. free abelian pro-p groups, are all finitely axiomatizable in the class of all profinite groups.
Finiteness of profinite groups with a rational probabilistic zeta function
证明了具有有理概率zeta函数的某类投射有限群仅具有n个极大子群。
The article deals with profinite groups in which the centralizers are abelian (CA-groups), that is, with profinite commutativity-transitive groups. It is shown that such groups are virtually pronilpotent. More precisely, let G be a profinite CA-group. It is shown that G has a normal open subgroup N which is either abelian or pro-p. Further, a rather detailed information about the finite quotient G/N is obtained.
Extensions of profinite duality groups
证明了若核满足一定的有限性条件，则该类有限对偶群在群扩张下是闭的。这推广了Pletch和Wingberg的早期结果。
Commutators and pronilpotent subgroups in profinite groups
设G是一个投射有限群，其中所有由算子生成的幂零子群都是周期的.我们证明了G'是局部有限的。
We present two uncountable families of finitely generated residually finite groups all having the same profinite completion. One consists of soluble groups, the other of branch groups.
In this article we calculate the profinite genus of the fundamental group of a $n$-dimensional compact flat manifold $X$ with holonomy group of prime order. As consequence, we prove that if $n\leq 21$, then $X$ is determined among all $n$-dimensional compact flat manifolds by the profinite completion of its fundamental group. Furthermore, we characterize the isomorphism class of profinite completion of the fundamental group of $X$ in terms of the representation genus of its holonomy group.
Finiteness properties of profinite groups
广义地说，群的有限性是具有有穷级的性质的任何推广。无限群理论的很大一部分是关于有限性以及它们之间的关系。投射有限群是这方面的一个重要例子，它是紧拓扑群，与它们的有限象有密切的联系。本文研究了profinite群可能具有的几个有限性之间的关系，以及有限群和投射有限群的结构。
Polycyclic groups and profinite isomorphism
针对几乎多循环群类，给出了判断两个群是否具有同构的投射有限完备化的算法。
The article deals with profinite groups in which the centralizers are pronilpotent (CN-groups). It is shown that such groups are virtually pronilpotent. More precisely, let G be a profinite CN-group, and let F be the maximal normal pronilpotent subgroup of G. It is shown that F is open and the structure of the finite quotient G/F is described in detail.
Profinite rigidity of affine Coxeter groups
本文证明了仿射Coxeter群在 profinite 意义下是刚性的。
It is proved that every finitely generated profinite group with fewer than $2^{\aleph_0}$ conjugacy classes of elements of infinite order is finite
Jordan-Holder Theorem for profinite groups and applications
本文将合成列与合成因子概念推广到投射有限群，证明其Jordan-Hölder定理，并应用于无限投射可解扩张的Galois定理，同时探讨非强完全投射有限群中抽象与拓扑合成因子的关系。
We construct a correspondence between the cohomology groups of a group $G$ relative to a family of subgroups $\famS$ and the classes of `relative extensions' of $G$ by abelian groups, modulo a certain equivalence relation. We establish this correspondence for both discrete and profinite group pairs. We go on to discuss the relationships of profinite group pairs of cohomological dimension one with free products and projective group pairs.
Krull-Schmidt Theorem for small profinite groups
本文证明小有限群满足 Krull-Schmidt 直积分解唯一性定理，并研究自由 pro-$\mathcal{C}$ 群的消去性质，给出小有限群的新判定准则。
We prove that a profinite group $G$ with positive rank gradient does not satisfy a group law. In the case when $G$ is a pro-$p$ group we show that $G$ contains a nonabelian dense free subgroup.
Pro-$\mathcal{C}$ groups acting on profinite trees
研究Pro - $\mathcal{C}$子群嵌入自由有限完备积的条件，在适当假设下推广到在有限完备树上$k$ - 无圈作用的有限完备群，且结果可应用于重要有限完备群类。
Presentations: from Kac-Moody groups to profinite and back
我们来回之间，一方面，介绍算术和卡茨-穆迪团体，另一方面，介绍了投射有限群，推导出一路上的新成果。
Profinite and Finite Groups Associated with Loop and Diffeomorphism Groups of Non-Archimedean Manifolds
研究了非阿基米德域上有限维空间上紧致流形的几何环和非同构群的$p$ -Adic紧化.引入了弱拓扑。研究了新构造的紧群的结构。讨论了这类投射有限群的表示。
A group G has restricted centralizers if for each g in G the centralizer C_G(g) either is finite or has finite index in G. A theorem of Shalev states that a profinite group with restricted centralizers is abelian-by-finite. In the present article we handle profinite groups with restricted centralizers of word-values. We show that if w is a multilinear commutator word and G a profinite group with restricted centralizers of w-values, then the verbal subgroup w(G) is abelian-by-finite.
On Profinite Groups of Type $\operatorname{FP}_\infty$
设$R$是一个profinite环.构造了一类大的投射有限群$\widehat{{\scriptstyle\bf L}'{\scriptstyle\bf H}_R}\mathfrak{F}$，包括$R$上所有可解的投射有限群和有限上同调维数的投射有限群.证明了若$G \in \widehat{{\scriptstyle\bf L}'{\scriptstyle\bf H}_R}\mathfrak{F}$为$\operatorname{FP}_\infty$ over $R$型，则存在$n$使得$H_R^n(G,R [[ G ]]) \neq 0$，并导出了$\mathbb{Z}_p$上$\operatorname{FP}_\infty$型的无挠可解pro-$p$群有有限秩，从而回答了Kropholler猜想的无挠情形.
Profinite Groups Associated to Sofic Shifts are Free
证明了Almeida与不可约sofic移位相关联的自由profinite半群的极大子群是自由投射有限群，推广了第二作者关于全移位情形的一个较早结果（其对应的极大子群是极小理想的极大子群）.对某些相对自由的profinite半群证明了相应的结果。我们还建立了自由profinite半群的核和与不可约sofic移位相关联的$\J$类之间的其他一些类比。
The generalised Fitting subgroup of a profinite group
有限群的广义Fitting子群是由所有幂零或拟单的次正规子群生成的群。这个子群在有限群理论中的重要性源于这样一个事实，即它总是包含自己的中心化器，因此任何有限群都是其广义Fitting子群的自同构自同构群的阿贝尔扩展。我们定义了一类推广这一现象的投射有限群，并探讨了投射有限群的结构的一些后果。
Algebraic properties of profinite groups
近年来，关于投射有限群的研究取得了许多进展。我们调查一些新的结果，并讨论开放的问题。一个中心主题是有限群分解成各种子集的有界积，从而产生拓扑群的代数性质。
Continuous cohomology and homology of profinite groups
我们发展上同调和同调理论的投射有限群$G$系数的Pontryagin对偶范畴的pro-discrete和ind-profinite $G$ -模，分别。标准的结果组（上）同调持有这个理论：我们证明版本的普遍系数定理，林登-Hochschild-塞尔谱序列和夏皮罗引理。
Commutators and commutator subgroups in profinite groups
设$G$是一个投射有限群.证明了换位子群$G'$是有限的原循环的当且仅当$G$的所有换位子的集合包含在可数个原循环子群的并中.
Profinite completion of Grigorchuk's group is not finitely presented
本文证明了Grigorchuk群$\mathcal{G}$的profinite完备化$\mathcal{\hat G}$不是一个投射有限群.通过证明$H^2(\mathcal{\hat G},\field{F}_2)$是无穷维的，我们得到了这个结果.证明了关于有限代数$\mathcal{G}/ St_{\mathcal{G}}(n)$的几个结果，包括极小表示和Schur乘子.
The class of multi-EGS groups is a generalisation of the well-known Grigorchuk-Gupta-Sidki (GGS-)groups. Here we classify branch multi-EGS groups with the congruence subgroup property and determine the profinite completion of all branch multi-EGS groups. Additionally our results show that branch multi-EGS groups are just infinite.
The article deals with profinite groups in which centralizers are virtually procyclic. Suppose that G is a profinite group such that the centralizer of every nontrivial element is virtually torsion-free while the centralizer of every element of infinite order is virtually procyclic. We show that G is either virtually pro-p for some prime p or virtually torsion-free procyclic. The same conclusion holds for profinite groups in which the centralizer of every nontrivial element is virtually procyclic; moreover, if G is not pro-p, then G has finite rank.
We investigate the ability of a free pro-$\CC$ group of infinite rank to abstractly solve abstract embedding problems, and conclude that for some varieties $\CC$, the profinite completion of any order, of a free pro-$\CC$ group of infinite rank, is a free pro-$\CC$ group as well, of the corresponding rank.
We introduce a class $\A$ of finitely generated residually finite accessible groups with some natural restriction on one-ended vertex groups in their JSJ-decompositions. We prove that the profinite completion of groups in $\A$ almost detects its JSJ-decomposition and compute the genus of free products of groups in $\A$.
We prove that the sign of the Euler characteristic of arithmetic groups with CSP is determined by the profinite completion. In contrast, we construct examples showing that this is not true for the Euler characteristic itself and that the sign of the Euler characteristic is not profinite among general residually finite groups of type $F$. Our methods imply similar results for $\ell^2$-torsion as well as a strong profiniteness statement for Novikov-Shubin invariants.
Higman-Thompson groups and profinite properties of right-angled Coxeter groups
本文证明了右角Coxeter群在Coxeter群中具有profinite刚性，并展示了某些RACG具有无限profinite genus，同时推广了Higman关于Thompson群$V$的生成结果。
Inverse system characterizations of the (hereditarily) just infinite property in profinite groups
给出了有限群的逆系统的极限恰好无穷或遗传恰好无穷的判别准则。更重要的是，这些标准是“普遍的”，因为所有（遗传）只是无限的投射有限群出现作为特定形式的限制。这是文章的更正和修订版本：“逆系统刻画的（遗传）只是无限性质的投射有限群”，公牛。LMS第44卷，3（2012）413- 425。
This article explores the interplay between the finite quotients of finitely generated residually finite groups and the concept of amenability. We construct a finitely generated, residually finite, amenable group $A$ and an uncountable family of finitely generated, residually finite non-amenable groups all of which are profinitely isomorphic to $A$. All of these groups are branch groups. Moreover, picking up Grothendieck's problem, the group $A$ embeds in these groups such that the inclusion induces an isomorphism of profinite completions. In addition, we review the concept of uniform amenabil…
Profinite completions of some groups acting on trees
研究了作用于树上的一类群的有限完备化问题。事实证明，对于一些考虑的群体，完成与树自同构的完全群中的群体的闭包重合。然而，我们引入了一个无穷级数的群体，这是不是这样，并描述了内核的自然同态的profinite完成到上述封闭的各个群体。
Words and pronilpotent subgroups in profinite groups
设$w$是一个多线性换位子字，即在$n$不同的群变量中具有权值$n$的换位子。证明了如果$G$是一个投射有限群，且其中所有由$w$ -值生成的幂零子群都是周期的，则其动词子群$w(G)$是局部有限的.
The structure of finite and locally finite groups in which every element has prime power order (CP-groups) is well known. In this paper we note that the combination of our earlier results with the available information on the structure of finite CP-groups yields a detailed description of profinite groups with that property. Then we deal with two generalizations of profinite CP-groups. Theorem 1.2. A profinite group G is virtually pro-p for some prime p if and only if for each nontrivial x in G there is a prime p (depending on x) such that the centralizer of x is virtually pro-p. Theorem 1.3. L…
We study the structure of profinite polyadic groups and we prove that a polyadic topological group $(G, f)$ is profinite, if and only if, it is compact, Hausdorff, totally disconnected. More generally, for a pseudo-variety (or a formation) of finite groups $\mathfrak{X}$, we define the class of $\mathfrak{X}$-polyadic groups, and we show that a polyadic group $(G, f)$ is pro-$\mathfrak{X}$, if and only if, it is compact, Hausdorff, totally disconnected and for every open congruence $R$, the quotient $(G/R, f_R)$ is $\mathfrak{X}$-polyadic.
On free profinite subgroups of free profinite monoids
我们回答了Margolis从1997年的一个问题，建立了一个自由生成的自由profinite幺半群的极小理想的极大子群是一个自由的投射有限群。更一般地说，如果$\mathbf H$是扩张闭的有限群簇，并且对于无限个素数$p$包含$\mathbb Z/p\mathbb Z$，则相应的结果对于自由pro- $\bar{\mathbf H}$幺半群成立。
On hereditarily just infinite profinite groups obtained via iterated wreath products
我们研究了一个推广的家庭非（几乎亲$p$）遗传只是无限投射有限群介绍J。\！S！威尔逊在2010年我们证明了这个族包含有限低秩的群。我们还表明，在这个家庭中的许多群体是不拓扑可表示的。
Frobenius subgroups of free profinite products
本文解决了Herfort和Ribes的一个公开问题：某些类型的Profinite弗罗贝尼乌斯群作为两个投射有限群的自由profinite积的闭子群出现.这也解决了一个问题的波普关于prossolvable子群的自由profinite产品。
We introduce various probablistic finiteness conditions for profinite groups related to positive finite generation (PFG). We investigate completed group rings which are PFG as modules, and use this to answer a question of Kionke and the second author on positively finitely related groups. Using the theory of projective covers, we define and characterise a probabilistic version of the $\mathrm{FP}_n$ property for profinite groups, called $\mathrm{PFP}_n$. Finally, we prove how these conditions are related to previously defined finiteness conditions and each other.
Bieri-Eckmann Criteria for Profinite Groups
在本文中，我们得到了profinite环$R$上的投射有限群和模是$\operatorname{FP}_n$型的充要同调和上同调条件，类似于抽象群的Bieri-Eckmann准则.我们使用这些证明，类组的类型$\operatorname{FP}_n$是封闭的扩展下，由子群的类型$\operatorname{FP}_n$，适当的合并自由产品和适当的$\operatorname{HNN}$ -扩展，为每个$n$。我们表明，作为这一结果，有限秩的基本顺从的投射有限群是$\operatorname{FP}_\infty$型在所有profinite $R$。对于任何在子群、子群和扩张下闭的有限群类$\mathcal{C}$，我们也构造了每个$n$的$\operatorname{FP}_n$型而不是$\mathbb{Z}_{\hat{\mathcal{C}}}$上的$\operatorname{FP}_{n+1}$型的pro-$\mathcal{C}$群.最后，我们表明，自然模拟的通常条件测量时，亲$p$组的$\operatorname{FP}_n$型失败的一般投射有限群，回答在否定的profinite模拟的一个问题的Kropholler。
A group word $w$ is said to be strongly concise in a class $\mathscr C$ of profinite groups if, for any group $G$ in $\mathscr C$, either $w$ takes at least continuum values in $G$ or the verbal subgroup $w(G)$ is finite. It is conjectured that all words are strongly concise in the class of all profinite groups. Earlier Detomi, Klopsch, and Shumyatsky proved this conjecture for multilinear commutator words, as well as for some other particular words. They also proved that every group word is strongly concise in the class of nilpotent profinite groups. In the present paper we prove that for any…
In this article we study the extent to which an $n$-dimensional compact flat manifold with the cyclic holonomy group of square-free order may be distinguished by the finite quotients of its fundamental group. In particular, we display a formula for the cardinality of profinite genus of the fundamental group of an $n$-dimensional compact flat manifold with the cyclic holonomy group of square-free order.
Free subgroups of finitely generated free profinite groups
给出了自由投射有限群的子群的自由性的新的和改进的结果：有限字的正规闭包包含在基元中的子群是自由的;无限生成的非交换自由投射有限群的每个无穷指数子群包含在无限生成的自由投射有限子群中.这些结果相结合的扭曲的圈积的方法哈兰，观察紧群的行动，和一个秩计数参数，以证明一个猜想的Bary-Soroker，Fehm和Wiese，从而提供了一个相当普遍的充分条件的子群是自由profinite。由于我们的工作，我们能够解决的猜想Jarden的希尔伯特域所产生的扭点的阿贝尔簇。
The strong profinite genus of a finitely presented group can be infinite
我们构造了第一个例子的有限表示，剩余有限群，其中包含一个无穷序列的非同构有限表示的子群，使每个包含映射诱导同构的profinite完成。
On conciseness of words in profinite groups
设w为组词。证明了如果w在投射有限群G中只有可数个值，则动词子群w（G）是有限的.本文在w是多线性换位子字或字x^2或字[x^2，y]的情况下证实了这个猜想。
A countably based profinite group can be naturally seen as a metric space with respect to a given filtration, and thus, it has a well defined Hausdorff dimension function. Barnea and Shalev found a group theoretical expression for the Hausdorff dimension of the closed subgroups of a profinite group $G$, opening, in this way, a bunch of possibilities to explore. In this paper we generalize Barnea's and Shalev's result to arbitrary closed subsets of $G$.
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